where \(g_e\)  is a dimensionless coupling constant, which can
be computed in axion models, e.g. [1]}. It argued in
the context of the DFSZ model
[2]}, [3]}, [4]} that It can be given by a
relation of the form
\(g_{ae}=\frac{1}{3}\left(1-\frac{\tan ^2{\beta }}{1+\tan ^2{\beta }} \right),\,\tan {\beta }=\frac{\upsilon _2}{\upsilon _1}, \) 
